Finding point B for perfect sub

[Cj]

Why is point B not accounted for compensated income? If we chose all x, wouldn't x*=30?

[J]

Accounting for compensated income would be finding Point C (hence the income effect). We're looking for point B, which only accounts for the substitution effect. 

You're right, we would want to choose all X. But only after the income effect. Point B should give the same amount of utility at Point A. Because at the new prices, X has a better bang for buck, we want X instead of Y. 

Taking our utility function

• U(x,y) = 2x + 3y

Setting it equal to utility at point A

• Point A (0,10)
• U(0,10) = 30
  
• 2x + 3y = 30

We want 0 of Y, because X has the greater bang for buck. And setting Y to 0 gives us the amount of X that would give us the same amount of utility as point A. 

• 2x + 3(0) = 30
  
• 2x = 30
  
• x = 15
• Point B (15,0)
  

[Cj]

Don't we use the compensated income to find point B and compensated price for point C? 

[J]

I personally wouldn't think of it in those terms, but that's me. Compensated income is just a generalized term, the idea you need to get conceptually is that a change in price alters your purchasing power, no actual change in income is happening. I wouldn't have included such a term in that little guide there, you can easily derive how to do it without that idea in your head as long as you understand the general theory behind it.

Just look at the steps of how to solve the problem.

Find your original consumption bundle (Point A), and the consumption bundle for the new prices. (Point C)

Substitute the original bundle into the utility function.

Find out which of the goods has greater bang for buck at the new prices.

Find the bundle which prefers the good with the greater bang for buck that has the same amount of utility as your original consumption bundle. (Point B)

To find substitution effect, compute B - A

To find income effect, compute C - B

Total effect is the sum of both effects.

I hope this makes sense.

[Lindsay Appell]

The step by step process you described looks perfect, J! And Cj, the attached picture with descriptions of A, B, and C looks perfect too!

Let me just add a little clarification on compensated income! Thinking about it graphically, we know that A and B share the same utility level (because they are on the same IDC) and B and C share the same prices (because their budget constraints have the same slope). Thinking back to our introductory material on budget constraints, we know that the x1 and x2 intercepts are m / p1 and m / p2, respectively. We know that B's budget constraint and C's share the same prices, but we also know that they are not exactly the same line - B's budget constraint is parallel to C's. Since the prices are the same, the only thing that could cause this shift in the intercepts is changing the income. So at point B, instead of using the actual income, we use a "fake", compensated income to artificially shift the budget constraint until it's tangent to A's IDC. 

It seems paradoxical to change the income at Point B in order to keep ourselves at the same purchasing power. Remember though that the reason why we are doing this is to maintain the same level of utility, which is a more accurate measure of purchasing power than actual income is.

Hopefully that all makes sense! Let me know if I can clarify further!

-Lindsay